Generating functions of the first and second kind for a local canonical map

First-kind data. Let $n\ge 1$ and let $O_q,O_Q\subseteq {\mathbb{R}}^n$ be nonempty open sets (The metric topology: a set is open when every one of its points has a ball around it inside the set; closed means open complement). Let $S:O_q\times O_Q\to \mathbb{R}$ be of class $C^3$ ($C^k$ Euclidean maps and diffeomorphisms) and assume

$\det S_{qQ}(q,Q)\mathrm{\ne }0((q,Q)\in O_q\times O_Q),$

where $S_{qQ}(q,Q)$ is the $n\times n$ matrix with entries ${\mathrm{\partial }}^{2}S\mathrm{/}\mathrm{\partial }{q}^i\mathrm{\partial }{Q}^j(q,Q)$ (Directional derivatives and partial derivatives of a map $U\subseteq {\mathbb{R}}^m\to {\mathbb{R}}^n$, The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case). Identify $O_q\times O_Q$ with the phase-domain coordinates $(q,Q)$ of ${\mathbb{R}}^{2n}$ (A phase domain, its canonical one-form and its standard symplectic two-form) and form the $C^2$ map

$\mathrm{\Psi }:O_q\times O_Q⟶{\mathbb{R}}^{2n},\mathrm{\Psi }(q,Q):=(q,S_q(q,Q)),$

whose derivative is the block matrix $D\mathrm{\Psi }(q,Q)=\left(\begin{array}{cc}{I}_n& 0\\ S_{qq}(q,Q)& S_{qQ}(q,Q)\end{array}\right)$. A generating function of the first kind is such an $S$; the local map generated by $S$ is, on a neighbourhood of a point $(q_{\ast },Q_{\ast })$, the $C^2$ map $C:(q,p)\mapsto (Q,P)$ defined by

$p:=S_q(q,Q),P:=-S_Q(q,Q),$

solved as follows: since $\det S_{qQ}\mathrm{\ne }0$, the inverse function theorem (The Euclidean inverse function theorem) makes $\mathrm{\Psi }$ a local $C^2$ diffeomorphism, so $p=S_q(q,Q)$ determines a $C^2$ local inverse $Q=Q(q,p)$ on a neighbourhood of $\mathrm{\Psi }(q_{\ast },Q_{\ast })$; then $C(q,p):=(Q(q,p),-S_Q(q,Q(q,p)))$.

Second-kind data. Let $O_q,O_P\subseteq {\mathbb{R}}^n$ be nonempty open sets and let $S:O_q\times O_P\to \mathbb{R}$ be of class $C^3$ with $\det S_{qP}(q,P)\mathrm{\ne }0$ everywhere. A generating function of the second kind is such an $S$; the local map generated by $S$ is, near a point $(q_{\ast },P_{\ast })$, the $C^2$ map $C:(q,p)\mapsto (Q,P)$ defined by

$p:=S_q(q,P),Q:=S_P(q,P),$

where again the relation $p=S_q(q,P)$ is solved locally for $P=P(q,p)$ by the inverse function theorem applied to $(q,P)\mapsto (q,S_q(q,P))$, whose derivative is $\left(\begin{array}{cc}{I}_n& 0\\ S_{qq}& S_{qP}\end{array}\right)$.

Recorded identities. Wherever $S$ is of the first kind, the identity $pdq-PdQ=dS$ holds; wherever $S$ is of the second kind, the identity $pdq+QdP=dS$ holds, both as identities of $C^2$ coordinate one-forms on the $(q,Q)$- resp. $(q,P)$-space with $p,P,Q$ read as the displayed functions (Finite-regularity differential forms, Hamiltonian fields and canonical maps in phase charts). The mixed-derivative hypotheses $\det S_{qQ}\mathrm{\ne }0$ resp. $\det S_{qP}\mathrm{\ne }0$ are part of the data.